Lattice thermal resistivity of III–V compound alloysSadao Adachi Citation: Journal of Applied Physics 54, 1844 (1983); doi: 10.1063/1.332820 View online: http://dx.doi.org/10.1063/1.332820 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/54/4?ver=pdfcov Published by the AIP Publishing

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Lattice thermal resistivity of III-V compound alloys Sadao Adachi Musashino Electrical Communication Laboratory, Nippon Telegraph and Telephone Public Corporation, Musashino-shi, Tokyo 180, Japan

(Received 20 September 1982; accepted for publication 14 December 1982)

Lattice thermal resistivities of I1I-V compound alloys have been analyzed with a theoretical prediction based on a simplified model of the alloy-disorder scattering. The theoretical prediction shows a quite good agreement with the experimental data on various III-V ternary compounds. This model has also been applied to obtain the thermal resistivity of In I ~ x Gax Asy P I ~ y lattice­matched to InP for the composition range ofO<.v< 1.0. The result indicates that the thermal resistivity increases markedly with alloying and exhibits a maximum value of about 24 W- I deg cm at an alloying composition ofy~0.75.

PACS numbers: 65.40-j, 65.90.ti, 66.70. + f

I. INTRODUCTION

Investigation of the heat transport phenomena in solids is an old topic which arises in strong connection with the fundamental physical properties of the solids. I Lattice ther­mal conductivity, or thermal resistivity, results essentially from interactions between phonons and from the scattering ofphonons by crystal imperfections. Knowledge of the ther­mal conductivity of semiconductors forms an important part in the design of power-dissipating devices, such as diodes, transistors, and semiconductor lasers. The thermal-conduc­tivity value is also necessary in calculating the figure of merit for thermoelectric devices (e.g., Peltier devices).

111-V compounds are thought to be more suitable ma­terials to study some of the thermal-conductivity properties, since they offer a wide range oflattice and electrical proper­ties and can be obtained in highly pure form so that the intrinsic properties should be investigated. The thermal­conductivity properties have, thus, been studied intensively for many III-V compounds, including some of the ternary alloys. Reviews by Holland I and Maycock2 throughly cover important theoretical and experimental aspects of the ther­mal-conductivity properties of III-V compounds in detail and discuss the relevant literature. Various accurate data on a wider range of the elements and binary compounds up to date have led to an increased understanding of the thermal­conductivity phenomena. Semiconductor alloys are well suited for investigating the effects of imperfections on the lattice thermal conductivity. It is important to point out that when large numbers of foreign atoms are added to host lat­tice as in alloying, the thermal conductivity decreases signifi­cantly.

An exact calculation of lattice thermal conductivity is possible in principle, but lack of knowledge of various pa­rameters (e.g., anharmonic forces and lattice vibration spec­tra) and the difficulty of obtaining exact solution of phonon­phonon interactions are formidable barriers to progress. It is, thus, interesting to investigate the consequence of a sim­ple model which is more amenable to calculation.

Abeles3 has proposed a phenomenological model to an­alyze the thermal conductivity of semiconductor alloys. The thermal conductivity has been expressed in terms of the lat-

tice parameters and mean atomic weights of the alloy and its constituents. Agreement has been obtained between calcula­tion and published experimental data on Ge-Si and GaAs­InAs alloys. Abeles's model, however, requires various ma­terial parameters and adjustable constants to obtain the best fit between calculation and experiment. Because of this rea­son, a more simple and reliable model is thought to be need­ed in practical aspect. This is a motivation of the present work.

In Sec. II, a theoretical model proposed in this study is described. This model is based on an interpolation scheme, and the effects of compositional variations are properly tak­en into account in the model. The compositional variations bring bowing (nonlinearity) on the thermal conductivity through strain and mass point defects. In Sec. III, we com­pare the present model with published experimental data on I1I-V ternaries. The results show a good agreement between calculation and experiment. The Inl ~xGaxAsy PI ~y alloy system is thought to be promising materials for a variety of high-speed transistor (IC) and optoelectronic device applica­tions. Because of the lack of experimental data, the thermal conductivity of Inl ~ x Gax Asy P I ~ y alloy is also presented in Sec. III by applying the present model.

II. THEORETICAL MODEL

An interpolation scheme is known to be a useful tool for estimating various physical parameters of alloys. If one uses the linear interpolation scheme, ternary material paramater (T) can be derived from binary parameters (B 's) by

TAxB,_xc(x) = xBAC + (1 - X)BBC

-a' +b'x (1 )

for alloys of the form AxBI~xC, where a'-BAC and b '=(B AC - BBC). Some experimental data, however, de­viate largely from the linearity relation of Eq. (1), and have an approximately quadratic dependence on the mole frac­tion of one compound x. One can recognize from the litera­ture that the thermal conductivity of ternary material is just an extreme case. 1.2 The ternary material parameter, in such a

1844 J. Appl. Phys. 54 (4), April 1983 0021-8979/83/041844-05$02.40 © 1983 American Institute of Physics 1844

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case, can be very efficiently approximated by the relation­ship:

TAxB,_xdx) =xBAC + (l-x)BBc + CA_B x(l -x)

_a + bx + cx2, (2)

where a=BAC ' b =(BAC - BBC + CA_B ), and c = - CA_B •

C A-B is a contribution arising from the lattice disordee gener­ated in ternary Ax B, _ x C system by the random distribution of A and B atoms in one of the two sublattice sites. The coefficient c is called a bowing or nonlinear parameter. This can be estimated by taking into account the virtual-crystal and disorder-scattering effects and, then, can be given by a sum of the intrinsic bowing C; found in the virtual crystal approximation and the extrinsic bowing Ce due to the effect of aperiodicity.4 Recently, the autho~ has shown that the material parameters ofIII-V compound alloys, such as lat­tice and dielectric constants, vary almost linearly with the

mole fraction x, and therefore it can be concluded that Eq. (1) gives a good estimation for the relevant alloy-compound pa­rameters. The lattice thermal conductivity, on the other hand, exhibits a strong quadratic-like dependence on the mole composition x, suggesting relatively large value of the bowing c in Eq. (2).

Abeles3 has done calculations using an analysis of the lattice thermal conductivity which was reasonably success­ful on semiconductor alloys. His theory is based on the mod­el developed by Klemens6 and Callaway,7 and starts from three kinds of relaxation times: TN 1 = B 10)2 (three-phonon normal process), Tv 1 = B20)2 (three-phonon umklapp pro­cess), and Tv 1 = AF0)4 (strain and mass point defect), where 0) is the phonon frequency, B I' B2 , and A are constants inde­pendent of 0). Fis the disorder parameter and is a function of the masses and radii of the constituent atoms [see Eq. (6)]. The thermal resistivity W(x) obtained by him is written as3

{tan-I U [1 - (tan-I U IUW }-I

W(x)IWp(x) = [1 + (5/9)a] U + [(1 +a)15a]U4-j U 2 _(tan- 1 UIU)+ 1 ' (3)

with

Uo(xf =AIF(x)Wp(X)-I,

U(X)2 = UO(X)2[ 1 + (5/9)a] -I.

(4)

(5)

In Eqs. (3H5), Wp(x) is the thermal resistivity of the crystal in which the disordered lattice is replaced by an ordered virtual crystal, a is a ratio of normal-process to umklapp­process scattering (i.e., a = B II B2 ), and A I is nearly constant within a group of the III-V compounds. F(x) can be ex­pressed by

F (x) = x( 1 - xl! [.:1M 1M (xW + E[.:10!8(xW J, (6)

where

.:1M=MA -MB' (7)

.:18 = 8A - 8B , (8)

M(x) =xMA + (l-x)MB, (9)

and

8 (x) = x8A + (1 - x)8B. (10)

In Eqs. (6)-(10),M; and 8; (i = A, B) are the masses and radii of the constituent atoms, respectively, and E is regarded as a phenomenological, adjustable parameter. The first and sec­ond terms in the square bracket ofEq. (6) correspond to the contribution from the mass-defect and strain scatterings, re­spectively. Abeles has concluded that the larger thermal re­sistivity of the GaAs-InAs alloy is predominantly due to the strain scattering and not the mass-defect scattering. We now show that our simple expression of Eq. (2) is essentially the same as that of Abeles's result.

As pointed out by Abeles,3 in the case of weak point­scattering (i.e., Uo< 1), Eq. (3) becomes

1845 J. Appl. Phys .. Vol. 54. No.4. April 1983

W(x) = Wp(x) + 3- 1(1 + 2a + E..a 2)

21

X [1 + (5/9)a]-2AIF(x)

=Wp(x) +A2F(x).

(11)

In the relatively strong point-defect scattering (Uo> 1), on the other hand, numerical calculation ofEq. (3) suggests that the value of W(x)lWp(x) increases almost linearly with increas­ing U 2

• This means that in the case of Uo> 1 Eq. (3)canalso be reduced to the form ofEq. (11). Note that Wp(x) in Eq. (11) defines the interpolation between the limits of compositional range (O..;x..; 1.0). Thus, it can be understood that Eq. (11) is expressed by the same form as Eq. (2). Under this considera­tion, the thermal resistivity of the ternary system AxB,_xC can be rewritten in the form:

W(x) = X WAC + (l-x)WBC + CA _B x(l-x). (12)

The same treatment can be used in the case of the quaternary system AI_xBxCyDI_y under consideration, being aware of two kinds of disorder effects: A-B disorder (CA _B ) due to the random distribution of the A and B atoms in the cationic sublattice, and C-D disorder (CC_D) due to the random dis­tribution of the C and D atoms in the anionic sublattice. The expression can, then, be given by

W(x,y) = (1 - x)yWAC + (1 - x)(l - y)WAD + xyWBC +x(l- y)WBD

+ CA _B x(l-x) + CC_Dy(l- y). (13)

Sadao Adachi 1845

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Gal In As -x x

------ EXPER.

= CALCU. 300 K

o~~~~~--~~~~~~

o GaAs

0.5

x 1.0 InAs

FIG. I. Thermal resistivity oflnAs-GaAs alloy as a function of composi­tion at 300 K. The solid line is a theoretical fit to the data using the present model [Eq. (12)]. The dotted line gives the calculated thermal resistivity using the model proposed by Abeles (Ref. 3).

III. ANALYSIS AND DISCUSSION

Figure 1 compares the calculated result ofEq. (12) (solid line) to the experimental data of InAs- GaAs alloy. The ex­perimental data are taken from Refs. 2 and 8 (dashed line). Table I lists the binary data used in the calculations. The theoretical curve obtained from Eq. (3) is also shown in the figure by dotted line. 3 The experimental data shows that the lattice thermal resistivity increases markedly with alloying and exhibits a maximum value of about 23 W- I deg cm at an alloying composition of x=O.5. This value is about ten times as large as those of related binaries ( W GaA. and W1nA.). Such a feature was motivated by desire to obtain increased perfor­mance for thermoelectric power conversion since the figure

TABLE I. Thermal resistivity Wof III-V binary compounds (300 K).

Material W (W- I deg cm)

GaAs 2.27-InAs 3.70" InP 1.47-GaP 1.30" AlAs I. lOb InSb 8.06c

GaSb 2.50d

-P. D. Maycock, Solid-State Electron. 10, 161 (1967). bM. A. Afromowitz, J. Appl. Phys. 44,1292 (1973). c G. Busch and E. Steigmeier, Helv. Phys. Acta 34, 1 (1961). dM. G. Holland, Phys. Rev.l34, A471 (1964).

1846 J. Appl. Phys., Vol. 54, No.4, April 1983

I I

~-~ , ... , , I ,

I , I ,

I ,

InAs P l-x x

------ EX PER. -- CALCU.

, \

\ \ \ \ \ \ \ \ \ \ \ \

\ \ \ \ \ \ ,

300 K

, ,

o~~~~~~~~~~~~ o 0.5

InAs x

1.0 InP

FIG. 2. Thermal resistivity of InAs-InP alloy as a function of composition at 300 K. The solid line is a theoretical fit to the data using the present model [Eq. (12)].

of merit for such device applications varies proportionally with W. It is clear from the figure that the present model shows an excellent agreement with the experimental data. The alloy-disorder bowing C1n-Ga is determined by this fit­ting to be about 72 W- I deg cm.

The lattice thermal resistivities of InAs-InP and GaAs-GaP ternary alloys are shown in Figs. 2 and 3, respec­tively. The theoretical curves are computed from Eq. (12). The experimental data are taken for InAs-InP alloy from Refs. 2 and 9 and for GaAs-GaP alloy from Refs. 2 and 10.

------ EXPER. -- CALCU.

300K

o~~~~ __ ~~~~~~~~ o

GaAs 0.5

x 1.0

GaP

FIG. 3. Thermal resistivity ofGaAs-GaP alloy as a function of composition at 300 K. The solid line is a theoretical fit to the data using the present model [Eq. (12)).

Sadao Adachi 1846

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These data also show maxima in W (x) at alloying composi­tions of about x = 0.5. The best fitting values of CA&-p 's are found to be 30 and 20 W- I deg cm for InAs-InP and GaAs­GaP alloys, respectively. These values are considerably smaller than that of C1n-Ga for InAs-GaAs alloy (72 W- I deg cm). The mass difference between the host lattice elements and added element causes anharmonicity, which increases the thermal resistivity. The mean atomic weight of In-Ga is about two times as heavy as that of As-P. This feature is thought to be connected with the marked differ­ence of values between C1n-Ga and C As-P'

Figure 4 compares the calculated result of Eq. (12) to the experimental data of GaAs-AIAs alloy. The data are from Afromowitz. II The best fit to the data is obtained with COa-AI = 30 W- I deg cm which is to be compared with the value of CAs-p~25 W- I deg cm for the InAs-InP and GaAs-GaP systems. This is thought to be a consequence of that the mean atomic weight of Ga-AI is nearly the same as that of As-P.

The alloy-disorder bowing CA_B in Eq. (2) can be char­acterized by the two alloyed elements A and B and not de­pend on the element C. The energy-gap bowing, for example, can be expressed only by the electronegativity difference between the two alloyed elements A and B.4 The model pro­posed in this study is also based on this picture, as seen from Eq. (12). It is interesting to point out that the value of C As-P

for InAs-InP alloy is nearly equal to that of C As-P for GaAs­GaP alloy (i.e., C As-P ~25 W- I deg cm). The agreement between experimental data and expression of Eq. (12) is ex­cellent and represents the successful explanation of the com­positional variation of the lattice thermal conductivity in the alloy systems.

14

12

~ 10 ~ u C> UJ 8 0 .. l-

t 6

~ 4

2

0 0

GaAs

• EXPER. --CALCU.

0.5

x

300 K

1.0 AlAs

FIG. 4. Thermal resistivity of GaAs-AIAs alloy as a function of composi­tion at 300 K. The solid line is a theoretical fit to the data using the present model [Eq. (12)].

1847 J. Appl. Phys., Vol. 54, No.4, April 1983

CALCULATED

OL-~~~~~~~~~~~ o 0.5 1.0

(InP) X (GaP) (InSb) (GaSb)

FIG. 5. Calculated thermal resistivities of InP-GaP and InSb--GaSb alloys as a function of composition at 300 K.

If the values of WAC' WBc,and CA_B are available, one can easily estimate the lattice thermal resistivity and its de­pendence on the mole fraction of some alloy system by using Eq. (12). Figure 5 shows, as examples, the lattice thermal resistivities of InP-GaP and InSb-GaSb alloy systems. The binary data used are listed in Table I. The alloy-disorder bowing of C1n-Ga = 72 W- I deg cm, determined on InAs­GaAs alloy, is assumed. This assumption is justified by the argument mentioned above. To our knowledge, there has been no report on these alloy systems. The calculated results show that W (x) exhibits maximum values at alloying compo­sitions of x~0.5 for InP-GaP and x~0.45 for InSb-GaSb alloy.

In I _ x Gax Asy PI _ y quaternary system has attracted great interest because it can be grown epitaxially on InP substrate without lattice mismatch over a wide range of com­positions covering a broad range of band gaps (0.75-1.35 eV)Y Inl_xGaxAsy PI_y/lnP lasers emitting in the 1.3-1.7 f.lm wavelength region become a promising candidate for the light source of an optical fiber communication system because of recent development of optical fibers in this spec­tral region. The thermal resistivity of Inl_xGaxAsy P I _ y alloy is an important parameter in the optimization of such devices because it influences both device operation and oper­ating life through its effect upon the active-layer tempera­ture. This property of Inl_XGaxAsy P I _ y alloys has not been determined until now. By using Eq. (13), we obtain the thermal resistivity of Inl _ x Ga" Asy PI _ y alloy over the en­tire range of compositions.

Figure 6 shows the three dimensional representation of the thermal resistivity ofInl _ x Gax Asy PI _ y quaternary al-

Sadao Adachi 1847

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2.27

GaAs

1.47

[?''-' "'" 'M'

FI G. 6. Three-dimensional representation of the thermal resistivity for In, _ x Gax Asy P, _ y quaternary alloy over the entire range of compositions.

loy. The comers of the figure correspond to the values of related binaries, as listed in Table I. The bowing parameters used are as follows: C'n-Ga = 72 W -) deg cm and C As--P

= 25W-) deg cm. The bold line in the figure is the locus of W (x, y) for compositions lattice-matched to InP. The lattice­matching relations between the compositions x and y can be expressed as5

x = 0.1894y (O,y,l.O) 0.4184 - 0.013y

(14)

on InP substrate. The thermal resistivity W(x, y) as a func­tion of the y-composition parameter for In) _ x Gax Asy p) _ y lattice-matched to InP is shown in Fig. 7. The composition parameters y = 0 and 1.0 correspond to InP and 100.53 Gao.47As, respectively. The calculated curve shows that the thermal resistivity increases markedly with alloying and ex­hibits a maximum value of about 24 W -) deg cm at an alloy­ing composition of y~O. 75. Heat generated in an In) _ x Gax

Asy p) _ y active layer must pass through substantial thick­nesses ofInP cladding and In) _ x Gax Asy p) _ y cap layers to reach the heat sink. The thermal resistance ofIn) _ x Gax Asy p) _ y IlnP heterostructure laser can be calculated using a

model of Joyce and Dixon13 in which two-dimensional heat flow is assumed to proceed from a uniformly excited stripe to a constant-temperature heat sink on one face. The calculated thermal resistivities may be subject to error but should be acceptable for thermal design consideration.

IV. CONCLUSION

Lattice thermal resistivities of III-V compound alloys have been analyzed with a theoretical prediction based on a simplified model of the alloy-disorder scattering. The theo­retical prediction requires one bowing parameter for ternary alloys and two bowing parameters for quaternary alloys, arising from the effects of alloy disorder. Agreement is ob­tained between the present model and published experimen-

1848 J. Appl. Phys .• Vol. 54. No.4. April 1983

CALCULATED

i' 20 U

~ In Go As P T

1-)( )( Y l-y ~

on InP !c{ ! ~

10

300 K

o~~~~~~~~~~~~. o 0.5 1.0

Y

FIG. 7. Thermal resistivity as a function of the y-composition parameter for In'_xGaxAsy P,_y lattice-matched to InP.

tal data on various III-V ternary compounds. The analysis has been applied to obtain the thermal resistivity ofIn) _ x

Gax Asy p) _ y lattice-matched to InP for the composition range of 0, y, 1.0. The result indicates that the thermal re­sistivity increases markedly with alloying and exhibits a maximum value of about 24 W-) deg cm at an alloying composition of y~O. 75.

ACKNOWLEDGMENTS

The author wishes to thank K. Oe, K. Kumabe, S. Ha­shimoto, and N. Kuroyanagi for their continual encourage­ment.

'M. G. Holland, in Semiconductors and Semimetals, edited by R. K. Wil­lardson and A. C. Beer (Academic, New York, 1967), Vol. 2.

2P. D. Maycock, Solid-State Electron. 10,161 (1967). 3B. Abeles, Phys. Rev. 131, 1906 (1963). 41. A. van Vechten and T. K. Bergstresser, Phys. Rev. B I, 3351 (1970). 's. Adachi, 1. Appl. Phys. 53, 5863 (1982). 6p. G. Klemens, Phys. Rev. 119, 507 (1960). 71. Callaway, Phys. Rev. 113,1046 (1959). 8M. S. Abrahams, R. Braunstein, and F. D. Rosi, 1 Phys. Chern. Solids 10, 204 (1959).

9R. Bowers, 1. E. Bauerle, and A. 1. Cornish, 1. Appl. Phys. 30, 1050 (1959). lOR. O. Carlson, G. A. Slack, and S. 1. Silverman, 1. Appl. Phys. 36, 505

(1965). liM. A. Afromowitz, 1. Appl. Phys. 44, 1292 (1973). I2R. E. Nahory, M. A. Pollack, W. D. lohnston, lr. and R. L. Barns, Appl.

Phys. Lett. 33, 659 (1978). I3W. B. loyce and R. W. Dixon, J. Appl. Phys. 46, 855 (1975).

Sadao Adachi 1848

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